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## Account of occasional wave breaking in numerical simulations of irregular water waves in the focus of the rogue wave problem

The issue of accounting of the wave breaking phenomenon in direct numerical simulations of oceanic waves is discussed. It is emphasized that this problem is crucial for the deterministic description of waves, and also for the dynamical calculation of extreme wave statistical characteristics, such as rogue wave height probability, asymmetry, etc. The conditions for accurate simulations of irregular steep waves within the High Order Spectral Method for the potential Euler equations are identified. Such non-dissipative simulations are considered as the reference when comparing with the simulations of occasionally breaking waves which use two kinds of wave breaking regularization. It is shown that the perturbations caused by the wave breaking attenuation may be noticeable within 20 min of the performed simulation of the wave evolution

The nonlinear modes of coherent structure development in the Atmospheric boundary layer are investigated. Two-scale model of Atmospheric boundary layer is used in the calculation. The velocity field splits into large-scale profile of horizontal wind velocity and threescale velocity field. The former depends only on the vertical coordinate. The latter is connected with roll circulation and is subject to the vertical coordinate and the coordinate perpendicular to the roll direction. The influence of turbulence is parameterized by turbulent viscosity. The modification of wind profile by rolls is taken into account. Depending on the Reynolds number, different types of the hydrodynamic instabilities specific to the Atmospheric boundary layer occurred. This appears at the relative orientation of the arising geostrophic wind and roll circulation, and also at the scales and space periods of the structures. As the Reynolds number grows, the mean energy and helicity increase. Within the range of the Reynolds number between 200÷300 the dependence is close to linear, which points to the possibility of utilizing weakly nonlinear theory methods, where perturbation amplitudes increase as Re1/2. The rise of the roll asymmetry followed by remarkable growing of the extreme amplitude of a longitudinal velocity component in the direction opposite to geostrophic wind compared to the amplitudes along the lines of geostrophic wind is detected. Increase of the positive component of helicity by contrast to the negative ones is observed simultaneously. A qualitative comparison between the modeling findings and the measured characteristics of the coherent structures observed in the Atmospheric boundary layer is carried out. In July 2007, these structures were measured by acoustic sounding methods in Kalmykia, where asymmetry in the distribution of longitudinal velocity component was observed as well. An apparent pattern of roll circulation begins to reproduce in the mesoscale atmospheric model RAMS under grid size about 500 meters. Reasonably correspondence between numerical simulation findings and observable vortex with centers lying about 1200÷1300 meters high is received. The values of turbulent viscosity and effective Reynolds number are typical for unstable stratification conditions.

A model that calculates the rates of cooling of liquid metal over the casting section atthe liquidus temperature by the finite difference method is created and programmed.By the cooling rates, the empirical formula determines the grain size. The following process parameters are taken into account: casting and molding materials, cast and mold dimensions, their initial temperatures, and the coefficient of contact thermal resistance.

The details of the numerical scheme and the method of specifying the initial conditions for the simulation of the irregular dynamics of soliton ensembles within the framework of equations of the Korteweg – de Vries type are given using the example of the modified Korteweg – de Vries equation with a focusing type of nonlinearity. The numerical algorithm is based on a pseudo-spectral method with implicit integration over time and uses the Crank-Nicholson scheme for improving the stability property. The aims of the research are to determine the relationship between the spectral composition of the waves (the Fourier spectrum or the spectrum of the associated scattering problem) and their probabilistic properties, to describe transient processes and the equilibrium states. The paper gives a qualitative description of the evolution of statistical characteristics for ensembles of solitons of the same and different polarities, obtained as a result of numerical simulations; the probability distributions for wave amplitudes are also provided. The results of test experiments on the collision of a large number of solitons are discussed: the choice of optimal conditions and the manifestation of numerical artifacts caused by insufficient accuracy of the discretization. The numerical scheme used turned out to be extremely suitable for the class of the problems studied, since it ensures good accuracy in describing collisions of solitons with a short computation time.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.